466 6 SEQUENCES, SERIES, AND PROBABILITY Section 6-3 Arithmetic and Geometric Sequences Arithmetic and Geometric Sequences nth-Term Formulas Sum Formulas for Finite Arithmetic Series Sum Formulas for Finite Geometric Series Sum Formula for Infinite Geometric Series For most sequences it is difficult to sum an arbitrary number of terms of the sequence without adding term by term. But particular types of sequences, arith- metic sequences and geometric sequences, have certain properties that lead to con- venient and useful formulas for the sums of the corresponding arithmetic series and geometric series. Arithmetic and Geometric Sequences The sequence 5, 7, 9, 11, 13,..., 5 ϩ 2(n Ϫ 1),..., where each term after the first is obtained by adding 2 to the preceding term, is an example of an arithmetic sequence. The sequence 5, 10, 20, 40, 80,..., 5 (2)nϪ1,..., where each term after the first is obtained by multiplying the preceding term by 2, is an example of a geometric sequence. ARITHMETIC SEQUENCE DEFINITION A sequence a , a , a ,..., a ,... 1 1 2 3 n is called an arithmetic sequence, or arithmetic progression, if there exists a constant d, called the common difference, such that Ϫ ϭ an anϪ1 d That is, ϭ ϩ Ͼ an anϪ1 d for every n 1 GEOMETRIC SEQUENCE DEFINITION A sequence a , a , a ,..., a ,... 2 1 2 3 n is called a geometric sequence, or geometric progression, if there exists a nonzero constant r, called the common ratio, such that 6-3 Arithmetic and Geometric Sequences 467 a n ϭ r DEFINITION anϪ1 2 That is, continued ϭ Ͼ an ranϪ1 for every n 1 Explore/Discuss (A) Graph the arithmetic sequence 5, 7, 9, .... Describe the graphs of all arithmetic sequences with common difference 2. (B) Graph the geometric sequence 5, 10, 20, .... 1 Describe the graphs of all geometric sequences with common ratio 2. EXAMPLE Recognizing Arithmetic and Geometric Sequences 1 Which of the following can be the first four terms of an arithmetic sequence? Of a geometric sequence? (A) 1, 2, 3, 5,... (B) Ϫ1, 3, Ϫ9, 27,... (C) 3, 3, 3, 3,... (D) 10, 8.5, 7, 5.5,... Solutions (A) Since 2 Ϫ 1 Þ 5 Ϫ 3, there is no common difference, so the sequence is 2 Þ 3 not an arithmetic sequence. Since 1 2, there is no common ratio, so the sequence is not geometric either. (B) The sequence is geometric with common ratio Ϫ3, but it is not arithmetic. (C) The sequence is arithmetic with common difference 0 and it is also geometric with common ratio 1. (D) The sequence is arithmetic with common difference Ϫ1.5, but it is not geometric. MATCHED PROBLEM Which of the following can be the first four terms of an arithmetic sequence? Of 1 a geometric sequence? (A) 8, 2, 0.5, 0.125,... (B) Ϫ7, Ϫ2, 3, 8,... (C) 1, 5, 25, 100,... nth-Term Formulas If {an} is an arithmetic sequence with common difference d, then ϭ ϩ a2 a1 d ϭ ϩ ϭ ϩ a3 a2 d a1 2d ϭ ϩ ϭ ϩ a4 a3 d a1 3d 468 6 SEQUENCES, SERIES, AND PROBABILITY This suggests Theorem 1, which can be proved by mathematical induction (see Problem 63 in Exercise 6-3). THE nTH TERM OF AN ARITHMETIC SEQUENCE THEOREM a ϭ a ϩ (n Ϫ 1)d for every n Ͼ 1 1 n 1 Similarly, if {an} is a geometric sequence with common ratio r, then ϭ a2 a1r ϭ ϭ 2 a3 a2r a1r ϭ ϭ 3 a4 a3r a1r This suggests Theorem 2, which can also be proved by mathematical induction (see Problem 69 in Exercise 6-3). THE nTH TERM OF A GEOMETRIC SEQUENCE THEOREM a ϭ a r nϪ1 for every n Ͼ 1 2 n 1 EXAMPLE Finding Terms in Arithmetic and Geometric Sequences 2 (A) If the first and tenth terms of an arithmetic sequence are 3 and 30, respec- tively, find the fiftieth term of the sequence. (B) If the first and tenth terms of a geometric sequence are 1 and 4, find the seventeenth term to three decimal places. ϭ Solutions (A) First use Theorem 1 with a1 3 and Now find a50: ϭ a10 30 to find d: ϭ ϩ Ϫ ϭ ϩ Ϫ an a1 (n 1)d a50 a1 (50 1)3 ؒ ϭ ϩ Ϫ ϭ ϩ a10 a1 (10 1)d 3 49 3 30 ϭ 3 ϩ 9d ϭ 150 d ϭ 3 ϭ ϭ ϭ (B) First let n 10, a1 1, a10 4 and use Theorem 2 to find r. ϭ nϪ1 an a1r 4 ϭ 1r10Ϫ1 r ϭ 41/9 6-3 Arithmetic and Geometric Sequences 469 Now use Theorem 2 again, this time with n ϭ 17. ϭ 17 ϭ 1/9 17 ϭ 17/9 ഠ a17 a1r 1 (4 ) 4 13.716 MATCHED PROBLEM (A) If the first and fifteenth terms of an arithmetic sequence are Ϫ5 and 23, 2 respectively, find the seventy-third term of the sequence. 1 1 1 (B) Find the eighth term of the geometric sequence , Ϫ , , . 64 32 16 Sum Formulas for Finite Arithmetic Series If a1, a2, a3,..., an is a finite arithmetic sequence, then the corresponding series ϩ ϩ ϩ ...ϩ a1 a2 a3 an is called an arithmetic series. We will derive two sim- ple and very useful formulas for the sum of an arithmetic series. Let d be the common difference of the arithmetic sequence a1, a2, a3,..., an and let Sn denote ϩ ϩ ϩ ...ϩ the sum of the series a1 a2 a3 an. Then ϭ ϩ ϩ ϩ ...ϩ ϩ Ϫ ϩ ϩ Ϫ Sn a1 (a1 d) [a1 (n 2)d] [a1 (n 1)d] Reversing the order of the sum, we obtain ϭ ϩ Ϫ ϩ ϩ Ϫ ϩ ...ϩ ϩ ϩ Sn [a1 (n 1)d] [a1 (n 2)d] (a1 d) a1 Adding the left sides of these two equations and corresponding elements of the right sides, we see that ϭ ϩ Ϫ ϩ ϩ Ϫ ϩ ...ϩ ϩ Ϫ 2Sn [2a1 (n 1)d] [2a1 (n 1)d] [2a1 (n 1)d] ϭ ϩ Ϫ n[2a1 (n 1)d] This can be restated as in Theorem 3. SUM OF AN ARITHMETIC SERIES—FIRST FORM THEOREM n S ϭ [2a ϩ (n Ϫ 1)d] 3 n 2 1 ϩ Ϫ By replacing a1 (n 1)d with an, we obtain a second useful formula for the sum. SUM OF AN ARITHMETIC SERIES—SECOND FORM THEOREM n S ϭ (a ϩ a ) 4 n 2 1 n 470 6 SEQUENCES, SERIES, AND PROBABILITY The proof of the first sum formula by mathematical induction is left as an exercise (see Problem 64 in Exercise 6-3). EXAMPLE Finding the Sum of an Arithmetic Series 3 Find the sum of the first 26 terms of an arithmetic series if the first term is Ϫ7 and d ϭ 3. ϭ ϭϪ ϭ Solution Let n 26, a1 7, d 3, and use Theorem 3. n S ϭ [2a ϩ (n Ϫ 1)d] n 2 1 ϭ 26 Ϫ ϩ Ϫ S26 2 [2( 7) (26 1)3] ϭ 793 MATCHED PROBLEM Find the sum of the first 52 terms of an arithmetic series if the first term is 23 3 and d ϭϪ2. EXAMPLE Finding the Sum of an Arithmetic Series 4 Find the sum of all the odd numbers between 51 and 99, inclusive. ϭ ϭ Solution First, use a1 51, an 99, and Now use Theorem 4 to find S25: Theorem 1 to find n: n a ϭ a ϩ (n Ϫ 1)dSϭ (a ϩ a ) n 1 n 2 1 n ϭ ϩ Ϫ ϭ 25 ϩ 99 51 (n 1)2 S25 2 (51 99) n ϭ 25 ϭ 1,875 MATCHED PROBLEM Find the sum of all the even numbers between Ϫ22 and 52, inclusive. 4 EXAMPLE Prize Money 5 A 16-team bowling league has $8,000 to be awarded as prize money. If the last-place team is awarded $275 in prize money and the award increases by the same amount for each successive finishing place, how much will the first- place team receive? 6-3 Arithmetic and Geometric Sequences 471 Solution If a1 is the award for the first-place team, a2 is the award for the second-place team, and so on, then the prize money awards form an arithmetic sequence with ϭ ϭ ϭ n 16, a16 275, and S16 8,000. Use Theorem 4 to find a1. n S ϭ (a ϩ a ) n 2 1 n ϭ 16 ϩ 8,000 2 (a1 275) ϭ a1 725 Thus, the first-place team receives $725. MATCHED PROBLEM Refer to Example 5. How much prize money is awarded to the second-place team? 5 Sum Formulas for Finite Geometric Series If a1, a2, a3,..., an is a finite geometric sequence, then the corresponding series ϩ ϩ ϩ ...ϩ a1 a2 a3 an is called a geometric series. As with arithmetic series, we can derive two simple and very useful formulas for the sum of a geometric series. Let r be the common ratio of the geometric sequence a1, a2, a3,..., an ϩ ϩ ϩ ...ϩ and let Sn denote the sum of the series a1 a2 a3 an. Then ϭ ϩ ϩ 2 ϩ 3 ϩ ...ϩ nϪ2 ϩ nϪ1 Sn a1 a1r a1r a1r a1r a1r Multiply both sides of this equation by r to obtain ϭ ϩ 2 ϩ 3 ϩ ...ϩ nϪ1 ϩ n rSn a1r a1r a1r a1r a1r Now subtract the left side of the second equation from the left side of the first, and the right side of the second equation from the right side of the first to obtain Ϫ ϭ Ϫ n Sn rSn a1 a1r Ϫ ϭ Ϫ n Sn(1 r) a1 a1r Thus, solving for Sn, we obtain the following formula for the sum of a geomet- ric series: SUM OF A GEOMETRIC SERIES—FIRST FORM THEOREM a Ϫ a rn S ϭ 1 1 r Þ 1 5 n 1 Ϫ r 472 6 SEQUENCES, SERIES, AND PROBABILITY ϭ nϪ1 ϭ n Since an a1r , or ran a1r , the sum formula also can be written in the following form: SUM OF A GEOMETRIC SERIES—SECOND FORM THEOREM a Ϫ ra S ϭ 1 n r Þ 1 6 n 1 Ϫ r The proof of the first sum formula (Theorem 5) by mathematical induction is left as an exercise (see Problem 70, Exercise 6-3).